Boundary 𝐶*-algebras for acylindrical groups

Abstract

Let Δ \Delta be an infinite, locally finite tree with more than two ends. Let Γ > Aut ⁡ ( Δ ) \Gamma >\operatorname {Aut}(\Delta ) be an acylindrical uniform lattice. Then the boundary algebra A Γ = C ( ∂ Δ ) ⋊ Γ \mathcal {A}_\Gamma = C(\partial \Delta )\rtimes \Gamma is a simple Cuntz-Krieger algebra whose K-theory is determined explicitly.